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Sagot :
To solve this problem, we'll simplify each given algebraic expression and then arrange them based on the coefficients of [tex]\( n^2 \)[/tex]. Here are the expressions given:
1. [tex]\(-5\left(n^3 - n^2 - 1\right) + n\left(n^2 - n\right)\)[/tex]
2. [tex]\(\left(n^2 - 1\right)(n + 2) - n^2(n - 3)\)[/tex]
3. [tex]\(n^2(n - 4) + 5n^3 - 6\)[/tex]
4. [tex]\(2n\left(n^2 - 2n - 1\right) + 3n^2\)[/tex]
### Simplifying Expression 1
[tex]\[ -5(n^3 - n^2 - 1) + n(n^2 - n) \][/tex]
First, distribute and expand both parts:
[tex]\[ -5(n^3 - n^2 - 1) = -5n^3 + 5n^2 + 5 \\ n(n^2 - n) = n^3 - n^2 \][/tex]
Now combine these results:
[tex]\[ -5n^3 + 5n^2 + 5 + n^3 - n^2 = -4n^3 + 4n^2 + 5 \][/tex]
### Simplifying Expression 2
[tex]\[ (n^2 - 1)(n + 2) - n^2(n - 3) \][/tex]
First, expand both parts:
[tex]\[ (n^2 - 1)(n + 2) = n^3 + 2n^2 - n - 2 \\ n^2(n - 3) = n^3 - 3n^2 \][/tex]
Now combine these results:
[tex]\[ (n^3 + 2n^2 - n - 2) - (n^3 - 3n^2) = n^3 + 2n^2 - n - 2 - n^3 + 3n^2 = 5n^2 - n - 2 \][/tex]
### Simplifying Expression 3
[tex]\[ n^2(n - 4) + 5n^3 - 6 \][/tex]
Expand and combine the results:
[tex]\[ n^2(n - 4) = n^3 - 4n^2 \\ \][/tex]
Combine:
[tex]\[ n^3 - 4n^2 + 5n^3 - 6 = 6n^3 - 4n^2 - 6 \][/tex]
### Simplifying Expression 4
[tex]\[ 2n(n^2 - 2n - 1) + 3n^2 \][/tex]
Expand and combine:
[tex]\[ 2n(n^2 - 2n - 1) = 2n^3 - 4n^2 - 2n \\ \][/tex]
Combine:
[tex]\[ 2n^3 - 4n^2 - 2n + 3n^2 = 2n^3 - n^2 - 2n \][/tex]
### Simplified Expressions
Now we have the simplified forms:
1. [tex]\(-4n^3 + 4n^2 + 5\)[/tex]
2. [tex]\(5n^2 - n - 2\)[/tex]
3. [tex]\(6n^3 - 4n^2 - 6\)[/tex]
4. [tex]\(2n^3 - n^2 - 2n\)[/tex]
### Arranging by Coefficient of [tex]\(n^2\)[/tex]
1. [tex]\(-4n^3 + 4n^2 + 5\)[/tex] (Coefficient of [tex]\(n^2 = 4\)[/tex])
2. [tex]\(5n^2 - n - 2\)[/tex] (Coefficient of [tex]\(n^2 = 5\)[/tex])
3. [tex]\(6n^3 - 4n^2 - 6\)[/tex] (Coefficient of [tex]\(n^2 = -4\)[/tex])
4. [tex]\(2n^3 - n^2 - 2n\)[/tex] (Coefficient of [tex]\(n^2 = -1\)[/tex])
Arranged in ascending order of their [tex]\(n^2\)[/tex] coefficients:
[tex]\[ 6n^3 - 4n^2 - 6 \quad (-4)\quad , \quad 2n^3 - n^2 - 2n \quad (-1)\quad , \quad -4n^3 + 4n^2 + 5 \quad (4)\quad , \quad 5n^2 - n - 2 \quad (5) \][/tex]
1. [tex]\(-5\left(n^3 - n^2 - 1\right) + n\left(n^2 - n\right)\)[/tex]
2. [tex]\(\left(n^2 - 1\right)(n + 2) - n^2(n - 3)\)[/tex]
3. [tex]\(n^2(n - 4) + 5n^3 - 6\)[/tex]
4. [tex]\(2n\left(n^2 - 2n - 1\right) + 3n^2\)[/tex]
### Simplifying Expression 1
[tex]\[ -5(n^3 - n^2 - 1) + n(n^2 - n) \][/tex]
First, distribute and expand both parts:
[tex]\[ -5(n^3 - n^2 - 1) = -5n^3 + 5n^2 + 5 \\ n(n^2 - n) = n^3 - n^2 \][/tex]
Now combine these results:
[tex]\[ -5n^3 + 5n^2 + 5 + n^3 - n^2 = -4n^3 + 4n^2 + 5 \][/tex]
### Simplifying Expression 2
[tex]\[ (n^2 - 1)(n + 2) - n^2(n - 3) \][/tex]
First, expand both parts:
[tex]\[ (n^2 - 1)(n + 2) = n^3 + 2n^2 - n - 2 \\ n^2(n - 3) = n^3 - 3n^2 \][/tex]
Now combine these results:
[tex]\[ (n^3 + 2n^2 - n - 2) - (n^3 - 3n^2) = n^3 + 2n^2 - n - 2 - n^3 + 3n^2 = 5n^2 - n - 2 \][/tex]
### Simplifying Expression 3
[tex]\[ n^2(n - 4) + 5n^3 - 6 \][/tex]
Expand and combine the results:
[tex]\[ n^2(n - 4) = n^3 - 4n^2 \\ \][/tex]
Combine:
[tex]\[ n^3 - 4n^2 + 5n^3 - 6 = 6n^3 - 4n^2 - 6 \][/tex]
### Simplifying Expression 4
[tex]\[ 2n(n^2 - 2n - 1) + 3n^2 \][/tex]
Expand and combine:
[tex]\[ 2n(n^2 - 2n - 1) = 2n^3 - 4n^2 - 2n \\ \][/tex]
Combine:
[tex]\[ 2n^3 - 4n^2 - 2n + 3n^2 = 2n^3 - n^2 - 2n \][/tex]
### Simplified Expressions
Now we have the simplified forms:
1. [tex]\(-4n^3 + 4n^2 + 5\)[/tex]
2. [tex]\(5n^2 - n - 2\)[/tex]
3. [tex]\(6n^3 - 4n^2 - 6\)[/tex]
4. [tex]\(2n^3 - n^2 - 2n\)[/tex]
### Arranging by Coefficient of [tex]\(n^2\)[/tex]
1. [tex]\(-4n^3 + 4n^2 + 5\)[/tex] (Coefficient of [tex]\(n^2 = 4\)[/tex])
2. [tex]\(5n^2 - n - 2\)[/tex] (Coefficient of [tex]\(n^2 = 5\)[/tex])
3. [tex]\(6n^3 - 4n^2 - 6\)[/tex] (Coefficient of [tex]\(n^2 = -4\)[/tex])
4. [tex]\(2n^3 - n^2 - 2n\)[/tex] (Coefficient of [tex]\(n^2 = -1\)[/tex])
Arranged in ascending order of their [tex]\(n^2\)[/tex] coefficients:
[tex]\[ 6n^3 - 4n^2 - 6 \quad (-4)\quad , \quad 2n^3 - n^2 - 2n \quad (-1)\quad , \quad -4n^3 + 4n^2 + 5 \quad (4)\quad , \quad 5n^2 - n - 2 \quad (5) \][/tex]
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